Treffer: Partition Approximation via the Cube Root of Binomial-Partition Ratios: A First-Principles Derivation with Stirling-Based Correction

This work presents a novel approximation method for the integer partition function based on a cube root transformation of the ratio between central binomial coefficients and partition values. This paper provides a full theoretical foundation for this relationship, which the author first identified through empirical analysis in a previous technical note [here]. The method is derived rigorously from first principles using Stirling’s formula with Euler-Maclaurin corrections and the saddle point method applied to partition generating functions. A theoretically motivated, empirically optimized Stirling-based correction factor is introduced, yielding significant improvements in approximation accuracy across a wide range of input sizes. Extensive computational experiments demonstrate error reductions by up to 4.5× compared to classical Hardy-Ramanujan asymptotics, with stable and efficient performance validated up to n= 80,000. Key Contributions First-principles derivation of a cube root ratio-based approximation combining binomial coefficient and partition function asymptotics. Introduction of an adaptive correction factor α(n) = 3 + 1/(120n) motivated by Stirling’s approximation error terms. Empirical optimization of the correction coefficient that substantially improves accuracy across all tested ranges. Rigorous computational validation using Euler’s recurrence relation for exact partitions up to n=80,000. Demonstration of superior accuracy and numerical stability compared to classical Hardy-Ramanujan formulas. Insights suggesting deeper mathematical structure linking the adaptive correction to classical Stirling series that warrant further theoretical investigation. Included Files Venkat2025_PartitionApproximation.pdf: The full academic paper detailing the first-principles derivation, methodology, and performance analysis of the Stirling-corrected partition approximation formula. partition_forms_analyzer.py : Core Python script implementing the Stirling-corrected partition function approximation. ...